On the John-Strömberg-Torchinsky characterization of BMO
The John-Nirenberg definition of BMO sets the norm \[ \|f \|_{\text{BMO}} = \sup_Q \inf_c \left( \frac 1{|Q|} \int_Q |f(x) - c|\right) \] and it is known that the optimal choice of \(c\) is the median value of \(f\) on \(Q\) but the mean value \(f_Q\) works as well. Finiteness of the norm implies exponential decay of the distribution function of \(f - f_Q\), \[ |\{x \in Q||f(x) - f_Q|> t \}|\leq 2|Q|e^{-\frac{c(d) t}{\|f\|}}. \] The \text{BMO} norm condition is equivalent to the apparently weaker condition \[ \sup_Q \inf_{P \in \mathcal P_k} \left( \frac 1{|Q|} \int_Q |f(x) - P(x)|^p \right)^{1/p}. \] John gave another characterization of BMO that does not require local integrability: set \(\lambda_Q (E) = |E\cap Q|/|Q|\), and then if \(0 < s < 1, f\) measurable, \[ \|f\|_{\text{BMO}_s(R^d)} = \sup_Q \left( \inf_{c \in R} \left( \inf_{t \geq 0} \lambda _Q \{|f - c|>t \} < s \right) \right), \] and John proved that if \(0<s< 1/2\), \(\text{BMO}_s(R^d) = \text{BMO}(R^d)\). This is false for \(s > 1/2\) and Strömberg proved \(\text{BMO}_{1/2}(R^d) = \text{BMO}(R^d)\). It was pointed out by Shvartsman that \[ \|f\|_{\text{BMO}_s(R^d)} = \sup_Q \inf_c (f - c)^{*,Q}(s), \] where \(f^{*,Q}\) is the left-continuous nonincreasing rearrangement of \(f\) with respect to \(\lambda_Q\). The authors consider the replacement of the constant by polynomials. They say \(f \in \text{BMO}^k_s(R^d)\) if the norm below is finite: \[ \|f\|_{\text{BMO}^k_s(R^d)} = \sup_Q \inf_{P \in \mathcal P_k}(f - P)^{*,Q}(s), \] and prove that if \(0<s<1/2\), \[ \text{BMO}(R^d)/ \mathcal P_{k-1} = \text{BMO}^k_s(R^d), \] and that the result fails for \(s=1/2\) unless \(k = 1\). This improves the result of Strömberg-Torchinsky although Strömberg-Torchinsky also has a result in a general framework that includes these results when \(0<s<1/2\). However, the Strömberg-Torchinsky axiomatic construct is given for \(k = 1\) and does not make clear the difference between \(k = 1\) when the result holds for \(0 < s \leq 1/2\) and \(k>1\) where it holds for \(0 < s < 1/2\) and the range is maximal.
Strömberg-Torchinsky axiomatic construct
scientific article; zbMATH DE number 1269735